Math 615: Lecture of April 13, 2007
Note that given a nite ltration
0 = M0 M1 Mn1 Mn = M
of a nitely generated R-module M and an additive map L we have that
L(M ) = L(Mn /Mn1 ) + L(Mn1 ),
and, by induction on n, that
n
L(M ) =
L(Mj /Mj1 ).
j=1
In particul
Math 615: Lecture of April 6, 2007
We next want to prove unique factorization in all regular local rings, and we shall use
an entirely dierent method. We rst discuss the basic facts about the divisor class group
C (R) of a normal Noetherian domain R.
Prim
Math 615: Lecture of April 16, 2007
We next note the following fact:
Proposition. Let R be any ring and F = Rn a free module. If f1 , . . . , fn F generate
F , then f1 , . . . , fn is a free basis for F .
Proof. We have a surjection Rn
F that maps ei Rn t
Math 615: Lecture of March 23, 2007
Remark. It is worth noting that Cauchy sequences in an I-adic topology are much easier to
study, in some ways, than Cauchy sequences of, say, real numbers. In an I-adic topology,
for cfw_rn n to be a Cauchy sequence it
Math 615: Lecture of March 19, 2007
Colon-capturing in homomorphic images of Cohen-Macaulay rings
We will need the following two preliminary results:
Lemma (prime avoidance for cosets). Let S be any commutative ring, x S, I S
an ideal and P1 , . . . , Pk
Math 615: Lecture of March 16, 2007
We next want to study weakly F-rings, i.e., Noetherian rings of prime characteristic
p > 0 such that every ideal is tightly closed. Until further notice, all given rings R are
assumed to be Noetherian, of prime characte
Math 615: Lecture of April 2, 2007
We next prove that, up to non-unique isomorphism, a coecient ring of mixed characteristic p in which p is nilpotent is determined by its residue class eld and and characteristic
(the latter is a power of p). However, the
Math 615: Lecture of March 30, 2007
The following Theorem, which constructs coecient rings when the maximal ideal of
the ring is nilpotent, is the heart of the proof of the existence of coecient rings in complete mixed characteristic local rings. Before g
Math 615: Lecture of March 26, 2007
The results of the preceding Lecture imply that a complete local ring (R, m) that has
a coecient eld K is a homomorphic image of a formal power series ring in n variables
over K, where n is the least number of elements
Math 615: Lecture of March 21, 2007
We shall no longer be assuming that all rings have prime characteristic p > 0. Our
objective is to prove some basic results about the structure of complete local rings. We
shall begin by studying complete local rings th
Math 615: Lecture of April 4, 2007
Let p > 0 be a prime integer. We now know that a coecient ring of mixed characteristic
p and characteristic pn , where n 2, is determined up to isomorphism by its residue class
eld. Let K be a given eld of characteristic
Math 615: Lecture of March 28, 2007
Consider a complete local ring (R, m, K). If K has characteristic 0, then Z R K
is injective, and Z R. Moreover, no element of W = Z cfw_0 is in m, since no element of
W maps to 0 in R/m = K, and so every element of Z c